Estimating process batch flow times in a two-stage stochastic
flowshop with overlapping operations
I. Van Nieuwenhuyse • N. Vandaele
Department of Applied Economics, University of Antwerp, Prinsstraat 13, 2000 Antwerp, Belgium
[email protected] • [email protected]
This paper focuses on modelling the impact of lot splitting on average process batch flow times, in a
two-stage stochastic flowshop. It is shown that the traditional queueing methodology for estimating
flow times cannot be directly applied to a system with lot splitting, as the arrival process of sublots
at the second stage is not a renewal process. Consequently, an embedded queueing model is
developed in order to approximate the average flow time of the flags through the system; from the
flow time of the flags, the flow time of process batches can then be derived. The model turns out
to yield very satisfactory results, and provides a tool to quantify the reduction in flow time that
can be obtained by overlapping operations at different processing stages. Moreover, it allows to
model the trade-off between flow time improvement and gap time occurrence by using it within the
scope of a cost model.
1.
Introduction
The past decades have witnessed a surge in research efforts, aimed at modelling the performance of
stochastic production or service systems by means of queueing theory (e.g. Karmarkar et al. 1985,
Karmarkar 1987, Lambrecht et al. 1998).
While exact models are only available for a limited
number of settings (like M/M/1 or M/G/1, see e.g. Kleinrock 1975), approximative models have
been developed to estimate performance measures (like average flow times and WIP) under more
general conditions, such as GI/G/1 or GI/G/m (see e.g. Kramer and Lagenbach-Belz 1967, Whitt
1983, Whitt 1993). These models are highly valued for their speed and ease of use in providing
estimates of the performance indicators of interest, as opposed to e.g. simulation models (see Suri
et al. 1993, Suri 1998).
It is currently well-known that the average flow time of products through a system is influenced
by a range of managerial decisions, such as e.g. the product mix being produced in the shop,
the layout of the shop, and (last but not least) the batching policies used on the shopfloor. The
impact of batching policies on flow times is particularly interesting to analyze: by setting batch
sizes in a deliberate way, managers can in fact obtain flow time improvements without a radical
intervention in the system, and without large financial investments. Hence, setting batch sizes in
1
a production system is an important control (see e.g. Hopp and Spearman 2000, Lambrecht et al.
1998, Benjaafar 1996).
When studying the impact of batching policies, a distinction should be made between two types
of batches: i.e., process batches and transfer batches. A process batch (also referred to as a production batch or production lot) is defined as the quantity of a product processed on a machine
without interruption by other items (Kropp and Smunt 1990). In multiple-product environments,
the use of process batches is often unavoidable due to capacity considerations: to switch from one
product type to the next (e.g., to change fixtures or dies), a setup or changeover time is necessary,
which consumes part of the capacity of the machine. After a setup has been performed, a certain
quantity of the product (the process batch size) can be produced. Hence, a process batch can also
be defined as the quantity of a product produced between two consecutive setups.
The relationship between the process batch size and the average flow time has been thoroughly
studied, and turns out to be convex (Karmarkar 1987, Suri 1998). Current queueing models are
able to take into account the impact of the process batching policy (e.g. Karmarkar 1985a, 1985b),
and the insight into the convex relationship has stirred the development of optimization procedures, aimed at determining the optimal process batch size for a given objective function (e.g. the
weighted average flow time, Lambrecht et al. 1998).
The term transfer batch (also called transfer lot or sublot) on the other hand refers to the size
of a sublot of the process batch, moved after production on one machine to another operation or
machine (Kropp and Smunt 1990). The use of transfer batches is not caused by capacity considerations, but rather by flow considerations. Indeed, it is widely accepted that the use of transfer
batch sizes smaller than the process batch size can reduce product flow times by smoothing workflow and minimizing congestion levels (e.g. Santos and Magazine 1985, Benjaafar 1996, Goldratt
and Cox 1984, Hopp et al. 1990 and Umble and Srikanth 1995).
This is due to the mechanism
of overlapping operations: by allowing transportation of partial batches to a downstream station,
this station can already start processing these partial batches while work proceeds at the upstream
station, thereby accelerating the progress of work through the production facility (e.g. Graves and
Kostreva 1986, Jacobs and Bragg 1988, Litchfield and Narasimhan 2000).
Up to now, the impact of lot splitting on flow times in a stochastic setting has received relatively
little attention in the research literature.
There are a number of papers that have studied the
problem by means of discrete-event simulation (e.g. Wagner and Ragatz 1994, Smunt et al. 1996,
Jacobs and Bragg 1988, Ruben and Mahmoodi 1998). In the queueing literature however, the dis2
tinction between process batching and transfer batching is currently overlooked, as present models
assume that products move between the machines in process batch sizes (this is also referred to as
the lot-for-lot policy, see e.g. Van Nieuwenhuyse and Vandaele 2004a). To the best of our knowledge, only a few studies (Bozer and Kim 1996, Benjaafar 1996) have made an attempt to analyze
the impact of transfer batching on flow times analytically.
However, the assumptions made in
these papers are rather restrictive (e.g. Poisson arrival processes and/or Poisson service processes
at the different stages of the system, regardless of the batching decisions).
This apparent shortcoming of the current queueing models provided the basis for our research. In
this paper, we will study a two-stage, single-product flowshop, with a general arrival and service
process.
The application of the queueing methodology to this type of system entails a number
of difficulties, which, as will be shown, call for the development of an embedded queueing model.
This model turns out to yield very good estimations of the average process batch flow time, when
compared to simulation results.
Though the setting studied here is limited to two stages and one product type, we believe this
work provides an important first contribution, by presenting a generic methodology which, through
adequate modifications, may in the future be extended towards more general settings (such as
systems with multiple product types and job shop settings).
As a starting point, section 2 briefly describes the structure and assumptions of the two-stage,
single-product flowshop with overlapping operations.
It also outlines the difficulties arising in
modelling the impact of overlapping operations by means of the queueing methodology.
These
hurdles provide the argument for the development of the embedded queueing model, the framework
of which is described in detail in section 3. Section 4 then tests the performance of this embedded
queueing model for the two-stage stochastic system versus simulation results, and section 5 presents
a cost model to analyze the trade-off between flow time improvement and the occurrence of gap
times in a stochastic flowshop. Finally, section 6 summarizes the conclusions.
2.
The two-stage flowshop with overlapping operations
2.1
Structure and assumptions
The system we will consider is a two-stage, stochastic system with a single product type.
It is
assumed to be an open system: production orders are launched into the system, undergo operations
subsequently on the first and second stage, and leave the system after being processed. For the
3
sake of simplicity, we will assume that the size of a production order equals a process batch size.
The interarrival times of process batches in the system, as well as the setup and processing times
on the two servers, are generally distributed.
Both servers are assumed to be capacity servers
(implying that the processing time of a sublot is dependent upon the size of the sublot), and are
preceded by a buffer with infinite capacity and FIFO queueing discipline.
Figure 1 illustrates the progress of two consecutive process batches, consisting of 4 sublots each,
through the system.
Stage 1
Stage 2
Flow time
Setup
time
Flow time
Sublot processing
time
Gap time
Figure 1: Flowchart of 2 process batches, consisting of 4 sublots, going through a 2-stage stochastic
flowshop
Upon arrival, a process batch may have to wait in front of the first stage before being processed
(e.g., when this stage is still busy processing a previous process batch). When the server becomes
idle, the setup is performed and the product units in the process batch are processed one by one.
Products move between the two machines in transfer batches: it is assumed that the transfer batch
size is a common divisor of the process batch size, such that a process batch is split into an integer
number of sublots. As soon as the transfer batch size is reached, the sublot is moved to the second
stage (this is referred to in the literature as batch availability, e.g. Bukchin et al. 2002 and Santos
and Magazine 1985).
It is important to note that, in our setting, the setup time on stage two
can not start before the first transfer batch of the involved process batch has arrived in its input
buffer. This type of setup has been referred to in the literature as an attached setup (e.g. Chen
and Steiner 1998, Potts and Kovalyov 2000).
Hence, the first transfer batch of a process batch
acts as a flag (Smunt et al. 1996): its arrival in front of the second stage authorizes the start of the
setup, thereby causing the operations on stage two to partly overlap with the operations on stage
4
one.
Upon arrival at the second stage, the flag may have to wait until the server becomes idle. As soon
as it does, the setup for the process batch is performed and the flag is processed. The remaining
sublots are processed as soon as they are finished on stage one, provided that stage two is available
at that moment; otherwise, they will wait in the input buffer of the second stage.
The structure of the system described above implies that the second server may remain idle between
the processing of two consecutive sublots, belonging to the same process batch. These idle times
are referred to as gaps (Van Nieuwenhuyse 2004, Van Nieuwenhuyse and Vandaele 2004a).
As
illustrated in Figure 1, they occur whenever the second stage finishes processing a sublot before
the next sublot is available from the first stage. In deterministic settings, gaps can be avoided by
balancing the processing rates on the different machines in the shop (this is referred to as the noidling assumption, e.g. Baker and Jia 1993, Ramasesh et al. 2000); in stochastic settings however,
gaps may occur even when the system is perfectly balanced, due to the inherent variability in the
setup and processing times at the different stages.
The occurrence of gaps obviously leads to an increase in the average makespan of a process batch
on stage two, without adding value to the product. Nevertheless, as will become clear below, the
use of lot splitting will drive down average process batch flow times in our system, thanks to the
overlapping of operations.
2.2
Obstacles to the application of the queueing methodology
Current open queueing models analyze the performance of a production network by means of the
decomposition approach. In this approach, the network is decomposed into the different building
blocks (the servers).
Each server has its own specific arrival and processing characteristics.
In
a first step, the queue in front of each server is analyzed separately based on three different input
parameters: i.e., parameters for the arrival process, the service process and the utilization rate of
the server (see e.g. Whitt 1983, Whitt 1994, Buzacott and Shantikumar 1985, Suri et al. 1993
for more details). The different servers in the network are linked together by means of so-called
linking equations, which ensure that the output stream of a particular server represents the input
stream for the next server in the product’s routing. Finally, the results are recomposed in order
to determine the performance of the entire production system.
The decomposition approach is however based upon the critical assumption that the interarrival
times and service times at each service center are both independent and identically distributed
5
(IID), and that there is no correlation between interarrival and service times. Unfortunately, this
assumption does not hold in a system with lot splitting.
To substantiate this point, Figure 2 shows the arrival pattern for sublots at server 2, for the example
previously shown in Figure 1.
Stage 1
Stage 2
Arrival pattern of
sublots
Arrival pattern of flags
Figure 2: Flowchart with arrival pattern of sublots and flags
Typically, the arrival process will exhibit a bursty pattern: as soon as the flag has arrived, the remaining sublots of the same production batch will arrive within rather short time intervals. Next,
there will be a longer time interval before the next flag arrives. Consequently, the interarrival times
of sublots are correlated, and no longer stem from the same probability distribution. As mentioned
above, this violates one of the basic assumptions underlying the decomposition approach.
However, this issue may be circumvented by only considering the flags as entities moving through
the network, instead of looking at individual transfer batches. Figure 2 illustrates the argument:
indeed, the bursty pattern is caused by the fact that the interarrival times of the remaining sublots
do not follow the same probability distribution as the interarrival times of the flags.
When we
only consider the flags as entities moving through the network, and ignore the remaining transfer
batches, the bursty pattern disappears. The arrival process of flags in front of a machine can then
be approximated by a renewal process.
These observations lead us to the conclusion that, although the queueing methodology cannot be
applied to estimate the average flow time of individual sublots, it may be useful in modeling the
average flow time of the flags.
From the average flow time of the flags, we can then derive the
average flow time of a process batch. This is explained in detail in the next section.
6
3.
Embedded queueing model
In this section, we will describe the structure of the embedded queueing model. It is very similar to
the structure of the traditional open queueing models with a lot-for-lot policy ; the main difference
is that the level of analysis is not the process batches but the flags. The following notation will be
used:
m = machine number index (m = 1, ..., M )
N = process batch size
L = sublot size
T = number of sublots in a process batch (T =
N
L)
Yf,m = interarrival time of flags at stage m
YP B,m = interarrival time of process batches at stage m
λf,m = average arrival rate of flags at stage m
λP B,m = average arrival rate of process batches at stage m
SUm = setup time on stage m
xm = unit processing time on stage m
G2 = total gap time on stage 2
Wf,m = waiting time in queue of flags at stage m
Pm = process batch makespan at stage m
ρm = utilization of machine m
τf,m = interdeparture time of flags at machine m
FP B = flow time of a process batch through the flow shop
Ff = flow time of a flag through the flow shop
For any random variable Z, E(Z) will refer to the mean of the variable, s2Z to its variance, and c2Z
to its squared coefficient of variation (SCV).
3.1
Model structure and assumptions
As sublots are regrouped into process batches at the end of the routing, the average flow time of a
process batch going through the system can be approximated by estimating the average flow time
of the flag, and adding the average extra time elapsing on the second server before the completed
process batch leaves the system. This average extra time consists of the average total gap time
occurring on the second server, and the average processing time of the remaining T − 1 transfer
7
batches. This yields:
E(FP B ) = E(Ff ) + (T − 1) ∗ L ∗ E(x2 ) + E(G2 )
(1)
As mentioned above, the objective of the embedded queueing model is to yield a realistic estimate
of the E(Ff ).
This flow time consists of three components: the waiting time spent in queue at
both servers, the setup time experienced at both servers, and the processing time of the flag itself.
Hence:
E(Ff ) =
2
E(Wf,m ) + E(SUm ) + L ∗ E(xm )
(2)
m=1
Regarding each stage m as a G/G/1 server, we expect that the average waiting time of a flag in
front of each stage can be approximated by standard G/G/1 queueing expressions (such as the
Kraemer-Lagenbach-Belz expression, see Kraemer and Lagenbach-Belz 1967), which is in general
given by:
E(Wf,m ) =
⎧
⎪
⎨ =
ρ2m ∗(c2Y
⎪
⎩ =
ρ2m ∗(c2Y
f,m
+c2Pm )
2λf,m ∗(1−ρm )
f,m
exp{
+c2Pm )
−2∗(1−ρm )∗(1−c2Y
f,m
3∗ρm ∗(c2Y
+c2Pm )
f,m
)2
} c2Yf,m ≤ 1
c2Yf,m
2λf,m ∗(1−ρm )
(3)
>1
For the first stage, this yields no particular problems. As setup and processing times on a server
are assumed to be independent, we may write:
c2P1 =
s2SU1 + N ∗ s2x1
[E(SU1 ) + N ∗ E(x1 )]2
(4)
Moreover, as products arrive in process batches in front of stage 1, c2Yf,m will be equal to the SCV
of the interarrival times of process batches at stage 1, and hence is given:
c2Yf,1 = c2YP B,1
The utilization rate of stage 1 is given by:
ρ1 =
E(SU1 ) + N ∗ E(x1 )
E(YP B,1 )
Hence, E(Wf,1 ) can be readily calculated.
For the second stage however, the expressions for c2Yf,2 and c2P2 are more complicated: the interarrival
time of flags at the second stage (Yf,2 ) is determined by the interdeparture time of flags at the first
stage (which we will denote by τf,1 ), and the makespan of a process batch at the second stage (P2 )
needs to include the total gap time. Consequently, estimating E(Wf,2 ) first requires us to develop
appropriate approximations for these input parameters.
Approximations for τf,1 and P2 have been studied in detail in earlier research (Van Nieuwenhuyse
2004, Van Nieuwenhuyse and Vandaele 2004b, Van Nieuwenhuyse and Vandaele 2005). For ease of
reference, we highlight the most important results from these studies in the following subsection.
8
3.2
3.2.1
Approximating E(Wf,2 )
Approximation for c2Yf,2
From the structure of the system, it is clear that c2Yf,2 equals c2τf,1 :
c2Yf,2 = c2τf,1 =
s2τf,1
(5)
E[τf,1 ]2
In a stable system, the average interdeparture time of flags E[τf,1 ] will equal the average interarrival
time of flags E[Yf,1 ], and hence equals the average interarrival time of process batches in front of
stage 1 (E[YP B,1 ]):
E[τf,1 ] = E[Yf,1 ] = E[YP B,1 ]
(6)
As shown in Van Nieuwenhuyse and Vandaele (2004b), s2τf,1 can be approximated by:
s2τf,1
=
s2τP B,1
+ 2(1 −
T
ρ21 )Cov[
X1,i , R1 ]app
(7)
i=2
In this expression, s2τP B,1 refers to the variance of the interdeparture times of process batches under
a lot-for-lot policy. It is given by (see Marshall 1968):
s2τP B,1
= 2s2SU1 + N ∗ s2x1 + s2YP B,1
−2E[WP B,1 ] ∗ [E[YP B,1 ] − E[SU1 ] − N ∗ E[x1 ]]
(8)
The notation Cov[ Ti=2 X1,i , R1 ]app in expression (7) refers to the covariance between the processing
times of the remaining transfer batches on stage 1 ( Ti=2 X1,i ), and the idle time that may occur
afterwards, before the processing of the next production batch starts (this idle time is denoted by
R1 ).
This covariance is analytically intractable, but it has been shown (Van Nieuwenhuyse and
Vandaele 2004b) that it may be adequately approximated by rewriting:
T
X1,i = V
i=2
R1 = max[0, M ]
M = Yf,1 − SU1 − T ∗ X1
and assuming normal probability distributions for both V and M .
Note that this assumption
implies a zero-inflated normal probability distribution for R1 (see e.g. Blumenfeld 2001).
9
This
allows us to write:
Cov[ Ti=2 X1,i , R1 ]app
= E[V ∗ max(0, M )] − E[V ]E[max(0, M )]
+∞ +∞
= −∞ 0 v ∗ m ∗ fV,M (v, m) ∗ dmdv
(9)
E[M ]2
−(T − 1) ∗ E[X1 ] ∗
with
fV,M (v, m) =
2πsV sM
h(v, m)
= e
ρV,M
=
Exp{ 2 }∗s2M
2s
√ M
(
2ΠsM
1
+
E[M ]
2
E[M ]
∗ (1 + Erf [ √
]))
2s
M
∗ h(v, m)
1−ρ2V,M
(v−E[V ]) (m−E[M ]) (m−E[M ])2
(v−E[V ])2
−1
[
−2ρV,M
+
]
sV
sM
2(1−ρ2
)
s2
s2
V,M
V
M
(10)
−s2V
sV ∗sM
and
= (T − 1) ∗ E[X1 ]
E[V ]
E[M ] = E[Yf,1 ] − (E[SU1 ] + T ∗ E[X1 ])
s2V
= (T − 1) ∗ s2X1
s2M
= s2Yf,1 + s2SU1 + T ∗ s2X1
(11)
Note that the parameters used in calculating E[M ], s2M , E[V ] and s2V are all known exactly. The
notation Erf [x] in expression (9) refers to the error function in x:
x
2
exp{−t2 }dt
Erf [x] = √
π 0
(12)
Combining expressions (5), (6) and (7), we get the following approximation for c2τf,1 :
c2τf,1
=
c2τP B,1
2(1 − ρ21 )Cov[ Ti=2 X1,i , R1 ]app
+
E[Yf,1 ]2
(13)
This expression clearly reveals the relationship between c2τf,1 in case of a lot splitting policy, and
c2τP B,1 with a lot-for-lot policy.
3.2.2
Approximation for c2P2
From the definition of covariance, we know that:
c2P2 =
s2P2
E[P2 ]2
Morever, the random variable P2 is given by:
P2 = SU2 +
T
i=1
10
X2,i + G2
(14)
such that
E[P2 ] = E[SU2 ] + N ∗ E[x2 ] + E[G2 ]
and
s2P2
(15)
= s2SU2 + N ∗ s2x2 + s2G2
+2 ∗ Cov(SU2 + Ti=1 X2,i , G2 )
(16)
Hence, approximating c2P2 requires us to find expressions for E[G2 ], s2G2 and Cov(SU2 +
T
i=1 X2,i , G2 ).
Earlier research (Van Nieuwenhuyse and Vandaele 2005, Van Nieuwenhuyse 2004) has shown that
the total gap time G2 can be adequately approximated as follows:
G2 = max[Z, 0]
(17)
with Z a normally distributed random variable given by:
Z=
T
X1,k − (SU2 +
T
−1
X2,j )
(18)
j=1
k=2
This assumption implies a zero-inflated normal distribution for G2 , and yields the following approximation for E[G2 ]:
2
−E[Z]
2
E[Z]
E[Z] Exp{ 2s2Z } ∗ sZ
E[Z]
√
+
∗ Erf [ √
=
+
]
2
2
2πsZ
2sZ
E[G2 ]app
(19)
in which Erf [x] refers to the error function in x (see expression (12)), and E[Z] and s2Z are given
by:
E[Z] = (T − 1) ∗ [E[X1 ] − E[X2 ]] − E(SU2 )
s2Z
(20)
= (T − 1) ∗ [s2X1 + s2X2 ] + s2SU2
Using expression (15), this yields the following approximation for E[P2 ]:
E[P2 ]app = E[SU2 ] + N ∗ E[x2 ] + E[G2 ]app
(21)
The assumption of a zero-inflated normal distribution for G2 also allows us to write the following
approximation for s2G2 (Blumenfeld, 2001):
2
}∗
s2G2 ,app ≈ Exp{ −E[Z]
2s2
Z
−[
E[Z]∗s
√ Z
2π
−E[Z]2
Exp{
}∗s2Z
2s2
√ Z
2πsZ
+
E[Z]2 +s2Z
2
∗ (1 + Erf [ √E[Z]
])
2s
Z
(22)
+
E[Z]
2
11
∗ (1 +
E[Z]
Erf [ √
])]2
2sZ
Let’s now turn to Cov(SU2 +
T
i=1 X2,i , G2 ).
we can write:
Cov(SU2 +
T
It is clear that X2,T is independent of G2 , such that
X2,i , G2 ) = Cov(SU2 +
i=1
T
−1
X2,i , G2 )
i=1
Rewriting
T
−1
SU2 +
X2,i = U
i=1
and assuming a normal distribution for U (see Van Nieuwenhuyse 2004) allows us to write the
following approximation for Cov(SU2 + Ti=1 X2,i , G2 ):
Cov(SU2 +
T
i=1 X2,i , G2 )app
≈ E[U ∗ max[Z, 0]] − E[U ]E[max[Z, 0]]
+∞ +∞
≈ −∞ 0 u ∗ z ∗ fU,Z (u, z) ∗ dzdu
(23)
−E[Z]2
−(E[SU2 ] + (T − 1)E[X2 ]) ∗
Exp{
}∗s2Z
2s2
Z
√
(
2πsZ
+
E[Z]
2
∗ (1 + Erf [ √E[Z]
]))
2s
Z
with fU,Z (u, z) denoting the bivariate normal density function for the two correlated variables U
and Z:
fU,Z (u, z) =
h(u, z)
and with
2πsU sZ
= e
1−ρ2U,Z
∗ h(u, z)
2
])
−1
[ (u−E[U
2(1−ρ2
)
s2
U,Z
U
2
]) (z−E[Z]) (z−E[Z])
−2ρU,Z (u−E[U
+
]
2
s
s
U
Z
(24)
s
Z
E[U ] = E[SU2 ] + (T − 1)E[X2 ]
s2U
= s2SU2 + (T − 1) ∗ s2X2
ρU,Z
and E[Z] and
1
s2Z
=
(25)
−s2SU −(T −1)s2X
2
2
sU ∗sZ
as in expression (20).
Combining expressions (16), (22) and (23) then yields the following approximation for s2P2 :
s2P2,app = s2SU2 + N ∗ s2x2 + s2G2,app
+2 ∗ Cov(SU2 + Ti=1 X2,i , G2 )app
(26)
Using expressions (21) and (26), the approximation for c2P2 can then be written as:
c2P2,app =
s2P2,app
[E[P2,app ]2 ]
12
(27)
3.2.3
Resulting approximation for E[Wf,2 ]
Expressions (13) and (27) can now be plugged into expression (3) for evaluation of E[Wf,2 ]. The
utilization rate of stage 2 can be calculated in a straightforward manner from1 :
ρ2 =
4.
E(SU2 ) + N ∗ E(x2 )
E(YP B,2 )
Performance of the embedded queueing model
In this section, we will test the performance of the embedded queueing model in estimating E[FP B ]
versus results from discrete-event simulation.
To this end, we study two settings: in the first
setting (Case 1), the second stage has a considerably higher processing rate than the first stage,
whereas in the second setting (Case 2), the two stages are perfectly balanced. Consequently, the
gap times occurring in Case 2 are solely due to the variability present in the system.
4.1
Simulation experiment
Table 1 gives an overview of the input parameters for the two cases. Both cases where evaluated
at three different utilization rates for stage 1: ρM 1 = 30%, 60% and 90%. The random numbers
for interarrival times, setup and processing times in the simulation experiment were all drawn from
gamma distributions. This distribution was chosen because of its positive skewness, which is a
desirable characteristic for the individual time components. Other suitable candidates would have
been a lognormal distribution, or a beta(α1, α2) distribution with α1< α2. The runlength of the
simulation was equal to 100 000 process batches, of which the first 20 000 were considered to be
part of the warm-up period.
4.2
Results
Table 2 compares the resulting estimates for E[FP B ] for both Case 1 and Case 2, showing the
percentage deviation of the queueing model (E[FP B ]Q ) with respect to the simulation results
(E[FP B ]S ).
From the results, it appears that the embedded queueing model performs quite well: the percentage
deviations with respect to simulation are similar for the lot-for-lot policy (T = 1) and the different
1
Note that the average total gap time on the second server (E[G2 ]) need not be included in the utilization rate,
as it does not constitute a part of the time busy on server 2.
13
N
T
E[SU1 ]
E[SU2 ]
E[x1 ]
E[x2 ]
E[x1 ]
ρ1
c2SU1
c2SU2
c2x1
c2x1
c2YP B,1
Case 1
30
1,2,3,5,6,10,15,30
2
2
1
0.2
30%, 60% or 90%
0.75
0.75
0.5
0.5
0.75
Case 2
30
1,2,3,5,6,10,15,30
2
2
1
1
30%, 60% or 90%
0.75
0.75
0.5
0.5
0.75
Table 1: Input parameters for Case 1 and Case 2
T
1
2
3
5
6
10
15
30
E[FP B ]Q
E[FP B ]S
% dev
E[FP B ]Q
E[FP B ]S
% dev
E[FP B ]Q
E[FP B ]S
% dev
E[FP B ]Q
E[FP B ]S
% dev
E[FP B ]Q
E[FP B ]S
% dev
E[FP B ]Q
E[FP B ]S
% dev
E[FP B ]Q
E[FP B ]S
% dev
E[FP B ]Q
E[FP B ]S
% dev
ρ1 = 30%
44.7230
44.2365
(1.0998 %)
39.7231
39.2478
(1.2112 %)
38.7144
38.2441
(1.2296 %)
37.9134
37.4427
(1.2571 %)
37.7129
37.2431
(1.2616 %)
37.3127
36.8451
(1.2691 %)
37.1124
36.6472
(1.2696 %)
36.9123
36.4568
(1.2495 %)
Case 1
ρ1 = 60%
57.8358
56.7900
(1.8416 %)
52.8342
51.8012
(1.9941 %)
51.8220
50.7976
(2.0167 %)
51.0204
49.9962
(2.0486 %)
50.8196
49.7965
(2.0546 %)
50.4192
49.3985
(2.0662 %)
50.2188
49.2006
(2.0694 %)
50.0185
49.0102
(2.0573 %)
ρ1 = 90%
149.8667
142.4430
(5.2117 %)
144.8678
137.4543
(5.3934 %)
143.8667
136.4506
(5.4350 %)
143.0666
135.6492
(5.4681 %)
142.8666
135.4496
(5.4759 %)
142.4666
135.0516
(5.4905 %)
142.2666
134.8537
(5.4970 %)
142.0666
134.6633
(5.4977 %)
ρ1 = 30%
72.5929
68.9152
(5.3367 %)
58.4557
54.7106
(6.8452 %)
53.6230
50.0651
(7.1064 %)
49.7602
46.4214
(7.1924 %)
48.7898
45.5275
(7.1655 %)
46.8558
43.7802
(7.0251 %)
45.8864
42.9326
(6.8799 %)
44.9176
42.1144
(6.6563 %)
Case 2
ρ1 = 60%
91.6654
83.8548
(9.3145 %)
77.4342
69.5224
(11.3803 %)
72.5008
64.8216
(11.8467 %)
68.5973
61.1331
(12.2098 %)
67.6086
60.2357
(12.2401 %)
65.6592
58.4658
(12.3036 %)
64.6777
57.6051
(12.2776 %)
63.6998
56.7741
(12.1986 %)
Table 2: Results for E(FP B )Q and E(FP B )S , for Case 1 and Case 2
14
ρ1 = 90%
192.9906
185.3986
(4.0949 %)
178.5937
170.8449
(4.5355 %)
173.1278
166.0559
(4.2588 %)
169.0724
162.3155
(4.1628 %)
167.9903
161.3738
(4.1001 %)
165.9988
159.5666
(4.0310 %)
164.9665
158.6767
(3.9639 %)
163.9580
157.8308
(3.8821 %)
lot splitting policies (T > 1).
The performance is in general slightly worse for Case 2.
This is
not surprising: as the gap times in balanced lines are only due to the presence of variability, the
approximation for c2P2 tends to yield larger errors when compared to simulation results (see Van
Nieuwenhuyse 2004, Van Nieuwenhuyse and Vandaele 2005). The overall performance of the model
is nevertheless still satisfactory.
As shown in Figure 3, the curves for E[FP B ]Q and E[FP B ]S are convex and continuously decreasing
in the number of sublots, for both cases.
This convex behavior gives a graphical illustration of
the impact of overlapping operations, and confirms the general intuition that lot splitting has a
positive impact on average flow times, at least in a setting that consists of capacity servers.
In
settings that also contain delay servers, it should be noted that excessive use of lot splitting may
lead to capacity problems on these servers, as their utilization increases with the number of sublots
used. Hence, in these settings, the benefits of the overlapping operations effect may be countered
by increasing queueing times in front of the delay servers.
5.
Impact of lot splitting on total costs in a system with capacity
servers
In a two-stage stochastic flowshop with capacity servers, the use of lot splitting may or may not be
desirable depending on the relative importance of the inventory holding costs versus the gap costs.
Indeed, gaps may represent a cost for the production system because during gap times the server
has to be kept ”operational”, i.e. ready for processing the next transfer batch when it arrives;
depending upon the type of server, this may entail labour and/or energy costs. In this section, we
study the trade-off between these two cost types in more detail, and illustrate it by means of an
example.
5.1
Cost model and insights
Focusing on inventory holding costs and gap costs, the average total costs in a two-stage system
can be written as:
E[Costtot ] = E[CostInv ] + E[Costgap ]
From Little’s law (e.g. Kleinrock 1975), we know that the average work-in-process inventory in a
system is related to the average flow time through the system. Hence, in the stochastic system,
the average number of process batches (E[W IPP B ]) will be equal to the throughput rate of process
batches (λP B,1 ) times the average process batch flow time (E[FP B ]). This will be the case both
15
160
E[F ] ρ =0.3
PB Q 1
E[FPB]S ρ1=0.3
E[FPB]Q ρ1=0.6
E[FPB]S ρ1=0.6
E[FPB]Q ρ1=0.9
E[F ] ρ =0.9
140
PB S
120
1
100
80
60
40
20
0
0
5
10
15
20
25
30
T
200
E[F ] ρ =0.3
PB Q 1
E[FPB]S ρ1=0.3
E[FPB]Q ρ1=0.6
E[FPB]S ρ1=0.6
E[FPB]Q ρ1=0.9
E[FPB]S ρ1=0.9
180
160
140
120
100
80
60
40
20
0
0
5
10
15
20
25
30
T
Figure 3: E[FP B ]Q and E[FP B ]S in terms of T , for Case 1 (top pane), and Case 2 (bottom pane)
16
for a lot-for-lot policy and for a lot splitting policy:
E[W IPP B ]LF L = λP B,1 ∗ E[FP B ]LF L
E[W IPP B ]LS
= λP B,1 ∗ E[FP B ]LS
The average inventory holding costs (E[CostInv ]) per production cycle can then be calculated by
multiplying this average work-in-process inventory by a specified holding cost per process batch
per cycle, cInv (> 0):
E[CostInv ]LF L = cInv ∗ λP B,1 ∗ E[FP B ]LF L
E[CostInv ]LS
= cInv ∗ λP B,1 ∗ E[FP B ]LS
As, in a two-stage system with capacity servers, E[FP B ] is convex and continuously decreasing in
T , we can conclude that the following is valid:
E[W IPP B ]LF L > E[W IPP B ]LS
E[CostInv ]LF L > E[CostInv ]LS
The average inventory holding cost will be convex and continuously decreasing in T .
The average gap costs per production cycle can be determined by multiplying the average gap time
on the second stage (E[G2 ]) by a fixed cost cgap (> 0):
E[Costgap ] = cgap ∗ E[G2 ]
For the lot-for-lot policy (T = 1), the average gap costs are obviously equal to zero.
It can be
proven that E[G2 ] is concave and continuously increasing in T when E(x1 ) > E(x2 ) (see e.g. Van
Nieuwenhuyse and Vandaele 2005). When E(x1 ) < E(x2 ), the reverse may occur: E(G2 ) may be
decreasing in T .
Based upon the discussion above, we can conclude that the lot splitting policy will be preferable
to a lot-for-lot policy from a cost viewpoint when:
E[CostInv ]LF L > E[CostInv ]LS + E[Costgap ]LS
or, in other words, when the difference in inventory costs is not compensated by the occurrence of
the gap costs:
cInv ∗ λP B,1 ∗ (E[FP B ]LF L − E[FP B ]LS ) > cgap ∗ E[G2 ]
This will depend on the relative importance of cInv and cgap .
17
5.2
Example
As an illustration, we apply the model presented above to an unbalanced flowshop, for which the
parameters are given in Table 3. The arrival rate of process batches λP B,1 was chosen such that
ρ1 = 90%. The inventory holding cost cInv is equal to 10; the fixed gap cost cgap can either be low
(0.1), intermediate (0.15) or high (3).
E[xm ]
c2xm
E[SUm ]
c2SUm
N
m=1
0.5
1
2
0.75
30
m=2
0.1
1
2
0.75
30
Table 3: Input parameters for the example
Table 4 gives an overview of the resulting total costs for the three cases, in terms of the lot splitting
policy T . The optimal total costs are given in bold.
T
T
T
T
T
T
T
T
=1
=2
=3
=5
=6
= 10
= 15
= 30
cgap = 0.1
43.3090
41.9019
41.8010
41.7412
41.7273
41.7002
41.6868
41.6735
cgap = 0.15
43.3090
42.1055
42.1019
42.1215
42.1275
42.1403
42.1469
42.1536
cgap = 3
43.3090
53.7138
59.2554
63.7989
64.9406
67.2276
68.3724
69.5177
Table 4: Average total costs for the example, when cgap = 0.1,cgap = 0.15 and cgap = 3
As Table 4 shows, the optimal lot splitting policy from a cost perspective depends heavily on the
relative importance of the two components, i.e. the gap costs versus the inventory holding costs.
When the gap costs are very low (cgap = 0.1), the shape of the total cost curve is entirely dominated
by the shape of the average inventory holding costs; i.e., it is convex and continuously decreasing.
From a cost viewpoint, this implies that it is optimal to split the process batch in the maximum
number of transfer batches (Topt = N ). Conversely, when the gap costs are very high (cgap = 3),
the shape of the total cost curve is entirely dominated by the shape of the average gap costs; i.e.,
concave and continuously increasing as in our case E[x1 ] > E[x2 ].
From a cost viewpoint, it is
now optimal to use a lot-for-lot policy (Topt = 1).
Situations may also arise in which neither of the two components is clearly dominant at all values
18
of T . In that case, the optimal lot splitting policy will be located somewhere in between the two
extremes. For example, when cgap = 0.15, the optimum is located at Topt = 3.
6.
Conclusions
In this paper, we have studied how the queueing methodology can be applied to estimate average
process batch flow times in a stochastic environment with lot splitting.
The discussion has re-
vealed that we have to resort to an embedded queueing model, as the use of lot splitting entails
characteristics which prevent the direct application of queueing models to our type of system. The
objective of the embedded queueing model is to yield a good approximation of the average flow
time of the flags; from the average flow time of the flag, we can then derive the average flow time
of a process batch.
The use of the embedded queueing model was illustrated by means of two cases. The performance
of the model turns out to be very good when the two stages are not perfectly balanced.
balanced systems, the performance is slightly worse but still satisfactory.
For
The outcome of the
model confirms the general intuition that lot splitting has a positive impact on average flow time
in systems with capacity servers.
It is known that the use of lot splitting in a stochastic system automatically introduces the risk
of gap times occurring.
The presented model can be used as a basis for cost/benefit analysis:
hence, besides yielding good flow time approximations, it can provide added value in studying the
trade-off between flow time improvement and gap time occurrence.
Though the setting studied here is limited to two stages and one product type, we believe this work
provides an important first contribution, by presenting a methodology for studying the impact of
lot splitting in stochastic systems. The extension of the model towards more general settings, such
as systems with multiple product types and job shop settings, may require additional modifications.
Given the frequent use of lot splitting in real-life settings, these extensions undoubtedly provide an
interesting and challenging topic for further research.
Acknowledgments
This research was supported by the Research Foundation Flanders (Belgium). Dr. Van Nieuwenhuyse is currently a Postdoctoral Fellow of the Research Foundation Flanders.
19
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